Function interpolator

ABSTRACT

Methods and apparatus are disclosed for producing an output signal by interpolation from a finite plurality of samples representing values of a function at sample points thereof. The output signal is produced as the sum of reconstruction functions having similar shape, one each associated with each sample point and so chosen as to ensure a frequency spectrum that is substantially band-limited and an output signal that is substantially continuous when the samples are updated and to produce a constant output signal when the accepted samples are all equal.

Unite States Fatent 1191 [11] 3,35,16 Nathan Aug. 20, R974 FUNCTIONHNTERPOLATOR 3,480,767 11/1969 Howe"; 235/197 x 3,524,978 8/1970 Miuraet al. 235/197 X [76] Inventor- Nathan Habmshm 3,557,347 1/1971Robertson 235/197 x 1 Ha1fa, 34483, Israel [22] Filed: July 5, 1972Primary Examiner-Joseph F. Ruggiero [21] Appl. No.: 269,258 [57] ACTMethods and apparatus are disclosed for producing an Foreign Applicationfiy ata output signal by interpolation from a finite plurality of July7, 1971 Great Britain 31851/71 samples representing values of a functionat sample points thereof. The output signal is produced as the [52] US.Cl. 235/197, 235/150.53 sum of reconstruction functions having similarshape, [51] Int. Cl G06g 7/26 one each associated with each sample pointand so [58] Field of Search 1. 235/197, 150.53, 150.52, chosen as toensure a frequency spectrum that is sub- 235/150.5 1, 150.5, 194;340/347 AD, 347 PA stantially band-limited and an output signal that issubstantially continuous when the samples are updated [56] ReferencesCited and to produce a constant output signal when the ac- UNITED STATESPATENTS cepted Samples are all q 3,373,273 3/1968 Schubert 235/197 19Claims, 15 Drawing Figures C ON 7' FOL 5 TORA GE PATENTEU M162 0 19743., 831 9 Dig Slit! w 5 FUNCTION INTERPGLATOR The invention relates tointerpolators, or function generators, for a function of one or morevariables wherein a substantially continuous output is produced from afinite number of samples accepted at any one instant.

Let f(x) be a function of the variable x and denote the samples at x =iby f, =f(i), where i is an integer. Given M adjacent samples,f ,f andassuming f(x) to be sufficiently smooth function, in some domain ofxbetween x =j and x =j M I, it is required to produce an approximationg(x) to f(x) with the help of a sampling function s'"(x) such thatWhenever x traverses an integer value the process is updated, i.e., whenx increases and passes through an integer value, j is replaced by j j 1so that interpolation subsequently proceeds with the help of the samplesf f Similarly, when x decreases and passes through an integer value, jis replaced by j 1.

According to this invention, interpolation proceeds so that g(x) remainscontinuous during the updating process.

One prior method uses as sampling functions independently of the valueof M. Unless M is very large, this method has the disadvantage that theresulting function g(x) is far from band-limited. By band-limited ismeant the restriction of the spectrum GO) of g(x) to a finite frequencydomain, where f is the frequency and G) is the Fourier integral of g(x)defined by Therefore, even if f(x) is sufficiently closely bandlimited,g(x) does not, in general, provide a good reconstruction of f(x)according to this prior art.

The invention will now be set forth by way of example in the followingdescription taken in conjunction with the appended drawings in whichFIGS. l-3 are plots of sampling or reconstruction functions used incarrying out the invention;

FIG. 4 is a schematic diagram illustrating one of the ideas underlyingembodiments according to the invention;

FIG. 5 illustrates a device for producing the same function as that ofFIG. 4 in a different fashion;

FIG. 6 Is a schematic diagram relating to the prior art for theproduction of triangular input functions A (x) and A (x) which areuseful in connection with the invention, also shows an embodiment forthe conversion of the input signal into a digital and an analog partaccording to the prior art and finally shows how these signals can beapplied to the ideas of the invention;

FIG. 7 is a schematic diagram of an embodiment of the invention for M 4;

FIG. 8 is a schematic diagram of an embodiment of the invention for M2N;

FIG. 9 are plots of triangular functions A (x) and 20 FIG. 10 is aschematic diagram of an auxiliary device used in one embodiment of theinvention;

FIG. 11 relates to another embodiment of the invention for M 2N;

FIGS. 12 and 13 relate to the special cases of FIG. ll wherein N 2 and N3, resp.;

FIG. 14 relates to yet another embodiment of the invention for N 2; and

FIG. 15 is a schematic diagram of an embodiment of the invention forinterpolating a pulse train accepted in time sequence.

The spectrum of s(x) sin (rrx)/( n'x) is given by The functions s(x)used in the method according to the invention are defined by theirspectra S U) which are obtained from said SQ) by sampling it as follows:

l x tn= Z S(f)6(f= l-) where is a delta function, zero unless f O andsuch that f smomd 0 0 where G0) is any function that is continuous at f0; and the factor l/M is introduced as a convenient normalization. (Inthe above notation, the superscript M does not denote the M-th power.)Thus S U) is produced by uniformly sampling at intervals of UM a uniformband-limited spectrum. Sampling at jump discontinuities here meanstaking the average of the rightand left-hand values of the function atthe sampling point. The corresponding functions of x are as follows:

For odd values of M S (x) l/M [1 2 cos(21rx/M) 2 cos (41rx/M) ..+2 cos(mlynx/M],

For even values of M I write N M/2 and then These functions andapproximations thereto are used as sampling functions in the methodaccording to the invention.

The first few functions are therefore explicitly:

s (x) Va (1 2 cos(1rx/4) 2 cos2rrx/4) 2 cos(21rx/4) cosrrx) s (x) Va (12 cos(21rx/3)) s"(x) 1/5 (1 2 cos(21rx/5) 2 cos(21rx/5)) s (x) 1/7 (1 2cos(2'n'x/7) 2 cos(4rrx/7) 2 cos(61rx/7)) These sampling functions, foreven and odd M, resp., can also be transformed into the following moreconcise forms:

Marv-1 The last property causes the restauration of constants; i.e., ifall M samples in the sum for g(x) have the same value, then g(x) isconstant and equal to said value, in a suitable interval that isincluded in the sampled x-domain.

The above expressions for the sampling functions s(x) and s-(x) assumesample points at integer values of the variable x. They are, however,quite general, because the unit interval along x is arbitrary.Alternatively, taking a spacing of sampling points of X these samplingfunctions can be written as follows:

and g(x) becomes, for even and odd M, resp.,

it will be noticed that any such sampling function is a linearcombination of a constant and k sinusoidal functions, where k is thegreatest integer that is not greater than xM. Thus for even M, k N rM,and for odd M, k %(M- 1). Further, said k sinusoidal functions have alinear frequency progression. For s (x), for example, the basic angularfrequency is seen to be 7T/4, and the three further frequencies are2('rr/4), 3(11/4), and 4-(1r/4) 11'.

FIGS. 1, 2, and 3 are plots of the functions s (x), s (x), and s (x),resp.. The sampling functions are seen to assume their greatest value atx= 0. This property is general.

if the spectrum of/(x) is substantially band-limited to the frequencydomain to F,flx) is very closely reconstructed by g(x) provided that thesampling interval X is at most equal to about (M -2)/(2MF), which is,for even M, equal to (N l )/(2NF), and reconstruction of f(x) is stillquite good for values of X up to about (M O.35)/(2MF), or, for evenvalues of M, up to (N 0.7)/(2NF). It is here assumed that M is at leastequal to 3, and N is at least equal to 2.

For N =1, reconstruction is quite close as long as X (0.10)/F,approximately, and fair for 0.10/F E x 5 0.15/F.

it is advantageous to give yet another representation of the method ofinterpolation and updating used in the invention. Defining and so that t(x) represents the march of s-(x) for the duration of one period centredon x= 0 and vanishes outside this period, then where the samplinginterval X is taken as unity without loss of generality. The advantageof this representation lies in the fact that it includes updating in asmuch as it holds for all values of x, from minus to plus infinity. Thesummation is seen to be over all integer values of 1'. One samplingfunction (or interpolation function these expressions being heresynonymous) t-(x) is thus associated with each sample point but it isseen that at most 2N sample values affect g(x) for any given value of x,because the domain of t-(x) extends only over the values of x for whichN x 5 N. (Again, analogous results hold for odd values of M.)

Denoting by x j the greatest integer that is not greater than x, thelast formula can also be written in th f In the following examples ofthe invention these relations will be relied upon.

The examples relate mainly to hybrid (i.e., analogdigital) embodiments,but it will be quite clear therefrom how purely digital and purelyanalog embodiments can be implemented.

In hybrid embodiments, the variable x is decomposed into a digital partx and an analog part Ax, such that X x,,, "l' Ax where X [X] lt followsthat 0 Ax 1. if the implementations are shown to hold for [x] 0, theywill properly perform quite in general, provided only that updating ofthe f,

is taken care of. It is then only necessary to verify that theimplementations correspond to for even or odd values of M, resp.

FIG. 4 relates to the method of this invention for the case N =1. Ananalog signal x is first converted into the sum x x Ax where O 5 Ax land where x is in digital form and Ax is in analog form.-The prior artteaches how this can be achieved and will be described in some detaillater in this specification. V

Ax in the form of a voltage, is accepted by analog function generator 1having the transfer characteristic s,(y) for 0 y l, i.e., producing fromany input signal y the output signal For 0 x 1 there holds s (Ax)=s,(x). s,( x) is therefore produced at the output terminal of functiongenerator 1, because, in this instance, input signal y is equal to Axwhich is equal to x. Signal changing adder 2 accepts s (x) and 1 andproduces therefrom s (x) and s 1 Ax) are accepted by the analogterminals of the multiplying analog to digital converters (MDACs) 4 and5, respectively, and the values J", and f where j==x =x, are applied indigital form to the digital input means of MDACs 4 and 5, resp., fromcontrol and storage unit 3, and their outputs are added to produce g(x)in the x domain corresponding to x s x x -ll. MDACs are provided by theprior art. They have an input terminal for the acceptance of an analogsignal in the form of an electrical voltage (or current) and input meansfor the acceptance of a digital signal and produce at their outputterminal an analog signal representing the product of the two inputsignals. In the example of FIG. 4, the MDAC output is in the form of anelectrical voltage and the addition of the outputs of the two MDACS iscarried out in a non-sign changing adder, including an operationalamplifier, by wellknown prior art techniques. Control and storage unit 3is a digital computer and in itself not included in the in ventivematter of this invention. Function generator 1 is an analog functiongenerator whose input and output signals are in the form of electricalpotentials and whose transfer-characteristic is as specified above.

The device thus produces and as required, in the interval j x j +1 and,since j is any integer, g(x) is properly produced for any x becauseupdating is automatically taken care of in the device as described.

The device according to FIG. 5 uses analog function generator 10 toproduce from Ax as input the output cos(1rAx) MDAC 11 accepts at itsdigital input the value of /z(f, f to produce analog signal &0",- fcosh-A x). MDAC 12 need not be a multiplying digital-to-analog-converterbut merely a DAC (digital-to-analog-converter), its analog input being a(constant) reference voltage. 12 accepts the digital signal /z(f,+f toproduce analog V2(f;-l-f These are combined in adder 13 to produce theanalog signal 1 1 5( j f1+1) cos (11-Ax)+E (f 9 as required. Control andstorage unit 3 is a digital computer which controls the supply of thedigital signals.

In both examples, according to FIGS. 4 and 5, the digital control unitis responsive to the instantaneous value of x,, xj j. As long as x is inthe domain the value of j is fixed and the unit supplies f,- and f, inthe example of FIG. 4 and %(f f and /2(f;lf,- in the example of FIG. 5.If, now, x increases through x 1' and beyond, than j is replaced by j j+1. Similarly, if x decreases then as x goes through j, j is replaced byj' j 1. Similarly at all integer values of x. Thus updating is simplytaken care of by this method of implementing the invention.

One disadvantage of the methods according to FIGS. 4 and 5 is thediscontinuity of the analog signal Ax during updating. This can beovercome by the replacement of Ax by a saw-tooth or triangular input,such as that shown as A (x), FIG. 9. At the same time it is shown how toadapt the invention for use with nonuniform spacing of sample points.The sample points are now at values of x equal to x x x where x, xforj=. ,1,0,1, In the domain defined by x, g x 21 x there now hold x x,and the definition The functions A (x) and A (x) shown in FIG. 9 arepiecewise linear; they are linear between any two adjacent sample pointsand assume at adjacent sample points alternately the values 0 and 1.Thus the variation of A (x) and of A (x) is between 0 and 1. Moreover,A1(.x) A2(x) FIG. 6 is a schematic block diagram relating to theproduction of A (x) and A (x) from x as input, given in analog form. TheFIGURE also illustrates a method for the production of x and includesmeans for the production of the function g(x) for the case N 1 accordingto the relations given above.

The part of the FIGURE relating to production of x A (x), and A (x) isprior art and is taken from the following reference:

W. E. Chapelle, Hybrid Techniques for Analog Function Generation,Proceedings Joint Spring Computer Conference I963, pages 213-227. (Inparticular: Chapelles FIG. 7 which, as far as here relevant, is includedand reproduced in FIG. 6 of this specification.)

14 is a high-gain amplifier (the sign of its gain is appropriatelyswitched in order to assure stability of the device.) 15 is the countercontrol, 16 the counter, 17 the control logic, and 18 the storage unit.21 and 22 are MDACs receiving, from storage unit 18, x and x,,,,,,,,,respectively, in parallel binary digital form, and A,(x) and A (x),respectively, in analog form to produce A x and A x respectively, inanalog form. For x such that x, g x x x denotes x, or x if j orj l isodd, resp., and xeven denotes x, or x ifj orj l is even, resp.

A,x,,,, and A x are added and the result is differenced with x receivedin analog form, the resulting difference signal being fed to the inputof amplifier 14 to produce analog output signal A A (x). A A (x) isproduced therefrom in sign changing adder 19.

The feedback loop of amplifier 14 solves the equation x lodd zeven whereA l A,. Thus .X l odd l) even Therefore, solving for A,,

x even A1:A1(x) x0dd even odd even udd x even =l-Ax.

It follows that A,(x) and A (x) are indeed produced in this prior artcircuit in accordance with their definitions in FIG. 9.

The production of x,, x in digital form is likewise taught by the priorart. Counter 16 is stepped up or down by unity whenever, while xincreases or decreases, resp., either A (x) or A (x) pass through zero.This is effected through counter control 15 which includes comparatorcircuits to detect the passage of A (x) and A (x) through zero as wellas decision circuits to determine whether counter 16 is to beincremented or decremented. All relevant values of x, and f, are storedin storage means 18. Control logic 17 is activated by contents (j) ofcounter 16, where j is determined by the relation X] x x for theinstantaneous value of x; it assigns to x xeve" the values x x for oddj, resp., and ;+1, x for even j, resp.; it also identifies theassociated values f and f and feeds x x f and f in digital form to thedigital inputs of MDACs 21', 22', 22, and 21, resp.

f, here denotes f(x) at x =x f j denote f(x) for x x and x x resp.

A (x) is accepted by analog function generator having the transfercharacteristic y to s (y) and thus producing s (A (x)); s (A (x)) l s (A(x)) is produced therefrom at the output of a sign changing adderaccepting l and s (A (x)). s (A (x)) and s,(A (x)) are accepted at theanalog input terminals of MDACs 21 and 22. g(x) is produced from MDACoutputs f s(A (x)) (at the output terminal of MDAC 21) and f s(A (x))(at the output terminal of MDAC 22), by addition.

A description of the part of FIG. 6 relating to the prior art will befound in the reference paper.

A (x) 0 whenever x x; andj is an even integer.

A (x) 0 whenever x x; andj is an odd integer. A (x) 1 whenever x x; andjis an odd integer. A (x) l whenever x x, andj is an even integer.

A (x) can replace Ax.

Similarly, for x 2 x x x 2 x g x Ax l A (x) and therefore 1 Ax A (x);andforx x x ,x x 2x Ax l A,(x) and therefore 1 Ax A,(x).

FIG. 7 is an embodiment of the invention for N 2. x Ax +x as before. Axis accepted by analog function generators 30-32 producing at theiroutput terminals the signals s (Ax l), s (Ax) and s (l Ax), resp. Signals (2 Ax) is produced in this example at the output terminal of signchanger 33 according to the relation which follows from in view of therelation of symmetry s (x) =s (x). Each one of the function generators30-32 operates only over the domain 0 l of input signal Ax. Thus thetransfer characteristics of function generators 30, 31, 32 arerepresented by those parts of s (x), FIG. 2, from x= l tox=2;x=0tox=l;andx=-l tox=0 (the mirror image of the part from O to l resp.Production of s (2 Ax) could similarly be effected through the use of afurther analog function generator, having a transfer characteristicrepresented by s (x) in the domain x 2 to x -l and fed by Ax. 34-37 areMDACs accepting the analog outputs of 30-33 and the digital inputsrepresenting ff f f f resp. g(x) is produced by summing the outputs ofMDACs 3437. Alternatively, function generator 32 can be replaced byanother function generator having the same transfer characteristic asdoes function generator 31 but fed by input signal 1 Ax, and the analoginput signal to MDAC 37 can similarly be produced in a functiongenerator having the transfer characteristic of function generator 30and also fed by l Ax.

It will be quite clear how the method of FIG. 7 can be extended forembodiments of the invention corresponding to other values of N or of M.

FIG. 8 relates to a further embodiment of the invention. It usesapproximations to the sampling functions s-(x) in order to achievesimplifications of method and apparatus. In this example, 43 and 44 areanalog function generators having transfer characteristics Ax to s (Ax),for Ax 1. Sign changing adder 46 produces from Ax the signal I Ax andthe outputs of 43 and 44 are therefore s (Ax) and s-(l Ax), resp. Duringthe first computing interval x Ax and x,, j 0, and MDACs 40 and 41receive said analog inputs and digital signals f and f resp., to producef s (Ax) and f,s-(l Ax) which are received by adder E. The function s(x) is approximated for 1 lxl N by polynominal approximations. In thisexample, the approximations consist of one parabolic are between any twoadjacent sample points, such that s (i Ax) is replaced by where theconstants a, (which depend upon the value of N) are suitably chosen. Oneset of useful values for the a, is given by the relation a a a etc., arenegative and a a a etc. are positive.

Thus, approximately,

Sign changing adder 47 produces from Ax and -l as its inputs the output1 2Ax. Squarer 45 produces therefrom the signal (1 2Ax) which isaccepted by sign changing adder 48 to produce This signal is the analoginput of MDAC 42 whose digital input is which is produced by digitalcomputer means. The outputs of 40 and 41 are combined to produce anapproximation to g(x) for 0 x 1. Upon updating, the function samples J,are stepped up so that f f f are replaced by f ,f ,f resp.; subsequentlybyf f fl,

. resp., etc.

Alternatively, Ax in FIG. 8 is replaced by A (x) defined as Ax in aneven numbered interval and as l Ax in an odd numbered interval, where aneven or odd interval corresponds to even or odd values of j in theinequalityjx j+1 and Ax=x0j;OAx l. MDAC 40 now receives in successiveintervals the digital input signals f f f f,, f f etc., and MDAC 41receives successively f f f f f f etc.

Updating of the digital input to MDAC 42 is as before, i.e., f, isreplaced by f, for all integer values of j. This is so because, in thisexample,

for all values of x.

FIG. 10 is a schematic diagram of another circuit for the production ofthe parabolic function used above.

There holds the relation In FIG. 10, the signal 4(l Ax) is produced fromAx as input signal in sign changing adder 50 whose second input receivesthe signal representing 1 The output of 50 is multiplied in analogmultiplier 51 by Ax to produce the required function. A yet furtherembodiment performing the same function produces -(Ax) from Ax in a signchanging squarer and subsequently combines Ax and (Ax) Yet otherembodiments of the invention replace the said approximation withparabolic arcs by approximations with half wave sinusoids. Thus 1 (2Axl) is replaced by sin(1rAx).

The function S-(x) can also be conveniently approximated in the intervalx 0 1. One such method produces this function as the sum of (I x) and(s-(x) (l x)) where the latter function is approximated in an analogfunction generator for convex functions or by polynomial approximation;it is in general sufficient to use a polynomial of the third order.

FIG. 11 relates to a general N-th order interpolator using A (x) and A(x) as defined in connection with FIG. 9 as input signals. Only ifsampling is uniform, so that x, j, do the frequency spectrum relationsdescribed earlier hold. Nevertheless, even for nonuniform spacing thereresults a useful interpolator according to the method of the invention.

A (x) is accepted by analog function generators 52, 53, 54, 55 and A (x)is accepted by analog function generators 56, 57, 58, 59 which have thetransfer characteristics defined by y (as input) to (as outputs) resp.MDACs 60, 61, 62, 63, 64, 65, 66, 67 have digital input terminals at 68,69, 70, 71, 72, 73, 74, 75, resp. and produce one analog output signaleach. These outputs are summed in summing means (not shown in theFIGURE) to yield g0) In an even numbered interval such as x=x ..x ,x

there holds Ax 10 1 Ax 20 and in an odd numbered interval, such as X=.xx0, x1-

there holds Ax A (x), l Ax A,(x).

Interval j, where j is an integer, is here defined as x,

It follows that the digital inputs to said MDACs must be as follows,during interval j: If j is even: 60, 61, 62, 63; 67, 66, 65, 64 receiverespectively and if j is odd they receive, respectively fN+hfN+J1ifN+J2ifhi;

fif-rvum -mm +l+L which is the same sequence in reversed order. Thusstepping up by unity entails the replacement of cachf, by f and thereversal of order.

This scheduling of digital input signals is controlled by digitalcomputer means.

FIG. 12 relates to the special case N=2 corresponding to FIG. 11. Thescheduling of digital inputs in successive computing intervals isexplicitely noted in the FIGURE. For example,f ,f ,f are the digitalinputs to MDAC 86 during x =x x,, 1:, x x x resp., etc. FIG. 13similarly, relates to case N 3 of FIG. 11.

In FIGS. 11-13,s-(y+N 1), ,s (y+ 1), s (y 2), etc. define the transfercharacteristics of the corresponding analog function generators. Forexample, function generator 85 produces output signal s y 1) whenreceiving input signal y; since the input signal is, in this instance,equal to A (x), it produces the output signal s (A (x) I).

In all previous examples of the invention the MDACs can be replaced byDACs receiving digital signals in conjunction with analog multipliersmultiplying DAC outputs with the analog function generator outputs.

While the examples relate to hybrid analog-digital embodiments of theinvention, it is by no means restricted thereto. For example, theinvention can be carried out in purely analog or purely digital form.

Thus the functions A (x) and A (x) are readily produced by digital meansand the analog function generators are described for example inconnection with FIGS. 11-13 are readily replaced by digital computermeans. This is so, in part, as the produced functions can beapproximated with sufficient accuracy by polynomials of low order. Formany applications it is quite sufficient to use polynomials of orderthree, i.e., expressions of the form In fact, in these expressions a Ofor all function generators 52-59 with the exception of 55 and 59.

A still further embodiment of the invention will be described inconnection with FIG. 14. y is a secondary input signal defined in termsof the primary input signal .1: by

y cos rrx The example relates to N=2. The identity cos rrx 2 cos (711/2)1 yields the following expressions for s y is accepted by functiongenerators 100-103 having transfer characteristics corresponding with l0f ,f ,f resp.; etc. as indicated in FIG. 14. Thus each MDAC digitalinput is updated every fourth interval, when subscript j advances byfour (assuming increasing x). MDAC outputs are added in summer 108 toproduce g(x) as output. Function generators -103 are comprised ofdevices performing linear operations and square root extraction inaccordance with the above specified relations, in well-known prior artmanner.

Interpolation functions s-(x) can be expressed in terms of cos(7rx/N)with the help of the following identities:

cos 2/3 2 c05 3 1 cos 33 4 cos fl 3cos B cos 4B=8 cosfl-S cos B+1 Itfollows that These relations lead to yet another embodiment of theinvention which is particularly useful when variable x denotes time sothat samples fi are now received in a temporal succession. Assuming nowthat this is the case we shall therefore, in conformity with commonusage, replace x by time t. The invention provudes where a is apredetermined delay equal to at least N. We shall take, by way ofexample, a 3 N l. The example of FIG. 15 corresponds to N 2 and a delaya 3. The uniformly spaced sample train f f.;,, f

. is accepted at terminal 108. Hold circuit 109 includes two holddevices which accept fi for even and odd values of i, resp., and holdthem for the duration of two intervals, being updated thereafter. Thus ffor example, is held from t= i to t= i 2 and subsequently replaced by fDigital control unit 111 controls the sample holding and updatingoperations. In this example, sinusoidal wave generator 110 (anoscillater) serves as timer for the interpolator. Its negative topositive zero crossings are accepted by control unit 111 both to timesampling through connection 112 and a sampling device connected thereto(not shown in the FIGURE) and to time the operation of hold 109. Clockor timing circuit 110 produces the output voltage In other examples ofthe invention, the incoming pulse train is used to synchronizeoscillator 110 which is phase-locked thereto.

Four-channel analog multiplexer 114 distributes the contents of 109after a delay of one interval to appropriate hold circuits in thefour-hold-circuit device 113. Operation of 114 and 113 is controlled bycontrol unit 111. the schedule of operations is as listed in thefollowing TABLE I in which the contents of the two hold' circuits inhold 109 and the contents of the four hold circuits in hold 113 arelisted against time t. Subcolumns 1 and 2 for hold 109 and 1 through 4for hold 113 refer to the separate hold devices included in 109 and 113,resp.

Block 109 also contains a two-channel multiplexer controlled by 111 forscheduling the incoming signals alternately to its two separate holdcircuits.

For example, at t=1 hold circuit 1 of hold 109 transfers its contents tohold circuit 1 of hold 113 and hold circuit 2 of hold 109 is updated byreplacement of its previous contents f., by f,. At t 2, thecontents ofhold circuit 2 of hold 109 are transferred to hold circuit 2 of hold 113and hold circuit 1 of hold 109 is updated by substitution of f for f TheTABLE is readily completed for all values of t by noticing that hold 113holds any value for the duration of four intervals and is subsequentlyupdated so that any f, previously held is replaced by f What is requiredin this instance according to the previously given relation is ln FIG.15, 114 is a sign preserving scaling adder receiving as inputs l andsin(1r t/2) and producing their sum multiplied by /2 as output, i.e.,/(-l sin(1r t/2) which is accepted by analog multiplier 115 and thereinmultiplied by the output sin(1r t/2) of oscillator 110. The output ofmultiplier 115 thus represents s,(: l) in the form of an electricalvoltage. Analog multiplier 125 receives this voltage and multiplies itby the instantaneous contents of hold circuit No. 1 of hold 113, alsosupplied in the form of a voltage, to produce at the output terminal of125 a voltage representing at t 0 the signal 116 is a phase shifterresponsive to the output of oscillator and producing therefrom thevoltage sin(1r(t l)/2) from which there is produced the signal s (t) bymeans of scaling sign preserving adder 117, having a scaling factor of/2, and analog multiplier 118; s (t) is subsequently multiplied inanalog multiplier 126 by the instantaneous contents of hold circuit No.2 of hold 113 to produce at t= 0 the signal 119 is a sign changerproducing from oscillator output the voltage sin(7rt/2), i.e.,introducing a 180 phaseshift, and 122 is a sign changer connected to theoutput of phase shifter 1 16 and produces at its output terminal thevoltage sin(rr(t 3)/2). s (t l) and s (t 2) are produced at the outputsof multipliers 121 and 124 with the help of sign preserving scalingadders and 123, resp., in a manner similar to that of production of s (tl), and they are multiplied in analog multipliers 127 and 128, resp., bythe instantaneous contents of hold circuits Nos. 3 and 4, resp., of hold113 to yield f ,,s' (t l) andf s (t 2), resp., at t= 0. The expressionsfor the outputs of multipliers -128 that hold at t= 0 continue to holdthroughout 0 t 1. At t 1 there occurs updating according to TABLE I andagain at any other integer value of t. Adder 130 combines the fourmultiplier outputs to produce g(t 3) according to the relation notedbefore. Updating takes care of the correct production of g(z 3) for allvalues of I.

The circuits associated with 116 through 124 merely produce theinstantaneous value of s (t l), which is present at the output ofmultiplier 115, for an argument t shifted by one, two and threeintervals. Thus s (t) is equal to the value assumed by s (t 1) oneinterval earlier. Other devices of this invention use different methodsto produce s (t), s (t l), and s (t 2). In particular, storage means areused in order to introduce the required delays.

For values of N other than N 2, there exist analogous embodiments aswill be quite clear from the above illustrative example. Thus for N 3,among other obvious changes in the circuit, hold 113 includes six holddevices and in order to be quite explicit the scheduling of values heldwill be given for N 3 in TABLE 11, delay a of 4 intervals being used.

The structure of the TABLE for the general case will now be quite clear.One way of seeing this is as follows: the first row has 0, f f in thefirst three columns, resp. Subsequently there appear, in this order, f ff The first three columns are completed in an obvious manner. The last2N columns can be written down as follows: The first thereof begins, asnoted, with f Subsequently, below, there are 2N entries f then 2Nentries f2; then 2N entires f then f etc. The rest of the entries followby noting that, starting at any position withf say, and going one to theright and subsequently one position down, there is f As a furtherexample, let the samples f, be stored in digital form. Take e.g. N 3.From x, whether given in digital form or converted through analog todigital conversion from analog into digital form, digital computingmeans determine the function cos [1r(x i)/N] for the instantaneous valueof x and for i N 1 -2, and subsequently compute the value of theexpression Next, i is stepped up one to i 1 and the process is repeated.Adding the results yields The process is repeated again and again up toi N 3, inclusive, and the final result is The process can be simplifiedby first computing all required values of cos [1r(x i)/N]and storingthem. Thus,

and it is seen that the number of computed functions can be reduced.

In the above methods of interpolators according to the invention, ifthere is given a finite data field, Le, a finite number of samples J,1%, f f say, and s (x) is used as interpolating function (analogousrelations holding for odd values of M) and if g(x) is required at theedges of the data field, i.e., for x xor x x then the method cannot giveresults because, in the first instance f f are unknown, and in thesecond instance f f are unknown. This matter is readily dealt with bymeans of data completion. One such method usesfl f i 0 andfi =f i kAnother method simply sets all missing values equal to zero.

While the above descriptions are for funbsions of omd v[riable, it isevident that the method can also be applied for functions of two or morevariables, provided that the samples are given at the lattice points ofa rectangular or higher order cubic grid. in two dimensions, let x, y,denote the point for which x i and y j and let f be the associated valueof f(x,y), and g(x,y) the required approximation which is to bedetermined with the help of an interpolating device. Then the method isapplied first for a plurality of adjacent values of j and the givenvalue of x, and subsequently, from the values g(x,y,) thus determined,for the given value of y, in complete conformity with the teachings ofthe prior art applying to a plurality of one dimensional functiongenerators and their use for the generation of functions of two or morevariables.

If fixed spacing of samples of f(x) is used and the method is applied,and if additional function values, likewise equally spaced but withsmaller spacing, are known, then it is possible to apply the method onceagain in order to obtain an improved fine resolution interpolation. Forthis purpose, the difference of the function g(x) first computed andsaid additional samples is determined and subsequently the method isreapplied to these difference sample values and their (smaller) spacing,the result being added to g(x).

Although this invention has been described and illustrated in detail, itis to be clearly understood that this is by way of illustration andexample only and is not to be taken by way of limitation.

What I claim is:

l. A function interpolator producing an output signal in accordance witha desired function of an independent variable from a plurality (M) ofsamples of a function of said variable at corresponding sampling pointscomprising:

First circuit means for the production of at least one analog inputsignal representing a predetermined function of said variable;

Second circuit means for storing a plurality (M) of signalsrespresenting said plural samples, respectively;

Nonlinear circuit means connected to said first circuit means forproducing a plurality of second output signals each representing thevalue of a reconstruction function for the instantaneous value of saidinput signal, wherein each of said reconstruction functions represents apredetermined nonlinear function of said variable;

a plurality of multiplier means each having input means for receivingone of said second output signals and at least one of said samplerespresentative signals and output means for producing thereat a thirdoutput signal;

summing circuit means connected to said plural multiplier output meanssumming said plural third output signals for producing said generatoroutput signal.

2. The interpolator as defined in claim 1 wherein said predeterminednonlinear function represents a linear combination of (k) sinusoidalfunctions of said variable of predetermined relative phase, (k) beingthe greatest integer that is not greater than (/2M).

3. The interpolator as defined in claim 2 wherein (M) is an evenpositive integer, (N) is defined as (/zM), and said nonlinear functioncorresponds to (x) representing said variable.

4. The interpolator as defined in claim 2 wherein (M) is an odd integergreater than two and said nonlinear function corresponds to (x)representing said variable.

5. The interpolator as defined in claim 1 wherein said second circuitmeans include digital circuit means and said sample representativesignals are stored therein in digital form; and wherein said multipliermeans receive said sample representative signals in digital form.

6. The interpolator as defined in claim 5 further comprising:

Digital control means responsive to a signal representing apredetermined function of said independent variable for updating saidplural stored sample signals whenever said independent variabletraverses one of said sampling points 7. The interpolator as defined inclaim 5 wherein said second output signals are produced in analog formand said multiplier means are comprised of multiplying analog-to-digitalconverter (MDAC) means.

8. The interpolator as defined in claim 1 wherein said one analog signalproduced in said first circuit means is in the form of a sinusoidalfunction of said independent variable.

9. The interpolator as defined in claim 8 wherein said nonlinear circuitmeans include analog multiplier means for the provision of thenonlinearity thereof.

10. The interpolator as defined in claim 8 wherein said second circuitmeans comprise digitally controlled means for storing said plural storedsignals in analog form and updating said stored signals whenever saidindependent variable traverses a sampling point; each of said pluralmultiplier means being comprised of analog multiplier means.

11. The interpolator as defined in claim 1 wherein said nonlinearcircuit means include linear circuit means receiving all but one of saidplural second output signals for the production of a further secondoutput signal.

12. The interpolator as defined in claim 1 wherein said one analogsignal produced in said first circuit means is in the form of a sawtoothfunction, changing linearly as a function of said variable from a firstpreassigned constant value at any one sampling point to a secondpreassigned constant value at the adjacent sampling point and changingin value from one to the other of said first and second constant valueson traversal of a sampling point.

13. The interpolator as defined in claim 1 wherein said one analogsignal produced in said first circuit means is in the form of acontinuous triangular wave function of said variable, being a linearfunction of said variable between any two adjacent sampling points andalternately assuming one and another of two preassigned constant valuesat adjacent sampling points.

14. The interpolator as defined in claim 13 wherein said first circuitmeans comprise means for the production of a second triangular inputsignal such that the sum of said first and second triangular signals issubstantially independent of the value of said variable.

15. A nonlinear function generator responsive to a digital signal forthe production of an analog output signal as a digitally controllednonlinear function of an analog input signal comprising:

Nonlinear circuit means having input means for receiving the analoginput signal and output means for producing thereat a plurality ofanalog reconstruction signals each characterised by a nowhere linearfunction of said input signal;

Digitally controlled storage circuit means for receiving the digitalsignal, storing a plurality of signals representing function values, andissuing selected stored signals;

Multiplier means connected to said nonlinear circuit means and saidstorage circuit means for receiving and multiplying each of said pluralreconstruction signals and a respective stored and issued signal forproducing a plurality of product signals;

Summing circuit means connected to said multiplier means summing saidplural product signals to produce said analog output signal.

16. A nonlinear function interpolator responsive to a sample signalsequence for the production of an analog output signal representing aninterpolation function to the sample sequence comprising:

Digitally controlled storage circuit means for storing and sequentiallyissuing said sample signals; Signal generator means having output meansfor producing thereat, as a function of time, a plurality ofreconstruction function signals, wherein said reconstruction functionsare periodic in time and any one thereof represents any other thereofshifted by a predetermined time interval; Multiplier means connected tosaid signal generator output means and said storage circuit means forreceiving said plural reconstruction function signals and said issuedsignals for the production of a plurality of analog product signals;

Summing circuit means connected to said multiplier means summing saidplural product signals to produce said analog output signal.

17. The interpolator as defined in claim 16 wherein said signalgenerator means include delay circuit means receiving one of saidreconstruction function signals for the production of at least onefurther said reconstruction function signal.

18. The interpolator as defined in claim 16 wherein said signalgenerator means include means for the production of a sinusoidal signaland nonlinear circuit means receiving said sinusoidal signal for theproduction of said plural reconstruction function signals.

19. The interpolator as defined in claim 18 wherein said nonlinear meansare comprised of linear circuit means and plural analog multipliermeans.

Patent No. 3,831 ,016 Dated August 20, 1974 Inventor(s) Amos Nathan Page1 0f 5 It is certified that error appears in the above-identified patentand that said Letters Patent are hereby corrected as shown below:

In the drawing Sheet 1, delete "BY" at bottom of page.

Sheet 5, FIGURES 14 and 15, should appear as shown on the attachedsheet.

Column 1, line 10, between "be" and "sufficiently" insert Column 2 line25, "f i/M" should read f i/M Column 2, line 63, "2" should read 3Column 2 line 65, "Zflx/S" should read 4 T x/S Column 3, line 54, "i-j"should read i= Column 4, line 10, "x" should read X Column 5, line 30,"Signal" should read sign Column 5 line 39 "=x" should read [x] Column5, line 33, should read Column 6, line 49, "21" should read Column 8,line 54, "f

" should read fj UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION3,831 ,016 August 20 1974 Patent No. Dated Amos Nathan Page 2 0f 5Inventor(s) It is certified that error appears in the above-identifiedpatent and that said Letters Patent are hereby corrected as shown below:

Column 1, line 4, column 2, line 46 and column 3, lines 8,

16, 19, 20 and 22, "m" should read M all occurrences.

Column 3, lines 42 and 43, "MX" should read MX all occurrences Column 3,lines 44 and 45 "Nx" should read NX all occurrences.

Column 3, lines 6, l7 and 23, "11" should read N all occurrences.

"TTX/X" should read nx/X Column 3, line 45,

Patent No.

UNITED STATES PATENT OFFICE Dated August 4 Inventor(s) Column 9 line 10,"lxl" should read \Jd Amos Nathan p 3 f 5 It is certified that errorappears in the above-identified patent and that said Letters Patent arehereby corrected as shown below:

Column 9, line 29, "i-l" should read i=1 Column 9, line 55, "-0" shouldread Column Column Column Column Column Column Column Column ColumnColumn Column Column Column 10, line 51, "A 90 should read A2 a 11, line2, "f N j I" should read f j 11, line l2, "f should read f ll, line 22,"Hi" should read 1 12, line 8, 1181" should read f l 12 line 32 betweenand "1" insert 12, line 39, "provudes" should read produces 12, line 41,should read a all four occurrences 12, line 46, "3" should read lZ, line53, "l" should read i 12, line 58, "later" should read lator 15, line 5,"N-l" should read N+l 15, line 21, "f should read v-- 5 Q PatentNo,3,831,016 ated August 20, 1974 Inventor(s) Amos Nathan Page 4 Of 5 It iscertified that error appears in the aboveidentified patent Q and thatsaid Letters Patent are hereby corrected as shown below:

Column 15, line 21, "fi" should read f Column 15, line 21, "cos2("should read cos a Column 16 line 16 "funbsions of omd" should readfunctions of one Column 16, line 17, "vlriable' should read variableSigned and Sealed this thirtieth D f March 1976 [SEAL] a A ttest:

RUTH C. MASON C. MARSHALL DANN Alresling Officer Commissioner uj'larentsand Trademarks

1. A function interpolator producing an output Signal in accordance witha desired function of an independent variable from a plurality (M) ofsamples of a function of said variable at corresponding sampling pointscomprising: First circuit means for the production of at least oneanalog input signal representing a predetermined function of saidvariable; Second circuit means for storing a plurality (M) of signalsrespresenting said plural samples, respectively; Nonlinear circuit meansconnected to said first circuit means for producing a plurality ofsecond output signals each representing the value of a reconstructionfunction for the instantaneous value of said input signal, wherein eachof said reconstruction functions represents a predetermined nonlinearfunction of said variable; a plurality of multiplier means each havinginput means for receiving one of said second output signals and at leastone of said sample respresentative signals and output means forproducing thereat a third output signal; summing circuit means connectedto said plural multiplier output means summing said plural third outputsignals for producing said generator output signal.
 2. The interpolatoras defined in claim 1 wherein said predetermined nonlinear functionrepresents a linear combination of (k) sinusoidal functions of saidvariable of predetermined relative phase, (k) being the greatest integerthat is not greater than ( 1/2 M).
 3. The interpolator as defined inclaim 2 wherein (M) is an even positive integer, (N) is defined as ( 1/2M), and said nonlinear function corresponds to 1/2N ( 1 + 2 cos ( pi x/N) + 2 cos (2 pi x/N) + . . . + 2 cos (N -1) pi x/N + cos pi x), (x)representing said variable.
 4. The interpolator as defined in claim 2wherein (M) is an odd integer greater than two and said nonlinearfunction corresponds to 1/M ( 1 + 2 cos (2 pi x/M) + 2 cos (4 pi x /M) +. . . + 2 cos (M- 1) pi x/M ), (x) representing said variable.
 5. Theinterpolator as defined in claim 1 wherein said second circuit meansinclude digital circuit means and said sample representative signals arestored therein in digital form; and wherein said multiplier meansreceive said sample representative signals in digital form.
 6. Theinterpolator as defined in claim 5 further comprising: Digital controlmeans responsive to a signal representing a predetermined function ofsaid independent variable for updating said plural stored sample signalswhenever said independent variable traverses one of said sampling points7. The interpolator as defined in claim 5 wherein said second outputsignals are produced in analog form and said multiplier means arecomprised of multiplying analog-to-digital converter (MDAC) means. 8.The interpolator as defined in claim 1 wherein said one analog signalproduced in said first circuit means is in the form of a sinusoidalfunction of said independent variable.
 9. The interpolator as defined inclaim 8 wherein said nonlinear circuit means include analog multipliermeans for the provision of the nonlinearity thereof.
 10. Theinterpolator as defined in claim 8 wherein said second circuit meanscomprise digitally controlled means for storing said plural storedsignals in analog form and updating said stored signals whenever saidindependent variable traverses a sampling point; each of said pluralmultiplier means being comprised of analog multiplier means.
 11. Theinterpolator as defined in claim 1 wherein said nonlinear circuit meansinclude linear circuit means receiving all but one of said plural secondoutput signals for the production of a further second output signal. 12.The interpolator as defined in claim 1 wherein said one analog signalproduced in said first cirCuit means is in the form of a sawtoothfunction, changing linearly as a function of said variable from a firstpreassigned constant value at any one sampling point to a secondpreassigned constant value at the adjacent sampling point and changingin value from one to the other of said first and second constant valueson traversal of a sampling point.
 13. The interpolator as defined inclaim 1 wherein said one analog signal produced in said first circuitmeans is in the form of a continuous triangular wave function of saidvariable, being a linear function of said variable between any twoadjacent sampling points and alternately assuming one and another of twopreassigned constant values at adjacent sampling points.
 14. Theinterpolator as defined in claim 13 wherein said first circuit meanscomprise means for the production of a second triangular input signalsuch that the sum of said first and second triangular signals issubstantially independent of the value of said variable.
 15. A nonlinearfunction generator responsive to a digital signal for the production ofan analog output signal as a digitally controlled nonlinear function ofan analog input signal comprising: Nonlinear circuit means having inputmeans for receiving the analog input signal and output means forproducing thereat a plurality of analog reconstruction signals eachcharacterised by a nowhere linear function of said input signal;Digitally controlled storage circuit means for receiving the digitalsignal, storing a plurality of signals representing function values, andissuing selected stored signals; Multiplier means connected to saidnonlinear circuit means and said storage circuit means for receiving andmultiplying each of said plural reconstruction signals and a respectivestored and issued signal for producing a plurality of product signals;Summing circuit means connected to said multiplier means summing saidplural product signals to produce said analog output signal.
 16. Anonlinear function interpolator responsive to a sample signal sequencefor the production of an analog output signal representing aninterpolation function to the sample sequence comprising: Digitallycontrolled storage circuit means for storing and sequentially issuingsaid sample signals; Signal generator means having output means forproducing thereat, as a function of time, a plurality of reconstructionfunction signals, wherein said reconstruction functions are periodic intime and any one thereof represents any other thereof shifted by apredetermined time interval; Multiplier means connected to said signalgenerator output means and said storage circuit means for receiving saidplural reconstruction function signals and said issued signals for theproduction of a plurality of analog product signals; Summing circuitmeans connected to said multiplier means summing said plural productsignals to produce said analog output signal.
 17. The interpolator asdefined in claim 16 wherein said signal generator means include delaycircuit means receiving one of said reconstruction function signals forthe production of at least one further said reconstruction functionsignal.
 18. The interpolator as defined in claim 16 wherein said signalgenerator means include means for the production of a sinusoidal signaland nonlinear circuit means receiving said sinusoidal signal for theproduction of said plural reconstruction function signals.
 19. Theinterpolator as defined in claim 18 wherein said nonlinear means arecomprised of linear circuit means and plural analog multiplier means.